Non-integrability criterion for  homogeneous Hamiltonian systems via blowing-up technique of singularities

Mitsuru Shibayama

doi:10.3934/dcds.2015.35.3707

Article Contents

2015, Volume 35, Issue 8: 3707-3719. Doi: 10.3934/dcds.2015.35.3707

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Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

Mitsuru Shibayama^1,

1.
Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531

Received: March 31, 2014

Revised: November 30, 2014

Published: May 2017

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Abstract

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be taken from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.

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Mathematics Subject Classification: Primary: 37J30; Secondary: 70F16.

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References

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